# zbMATH — the first resource for mathematics

A Banach algebra with its applications over paths of bounded variation. (English) Zbl 1408.46046
Following the study initiated by the author [J. Funct. Spaces 2018, Article ID 9345126, 10 p. (2018; Zbl 1400.46042)] where he introduced certain Banach algebras which generalized Cameron-Storvick’s Banach algebra, he now considers two new Banach algebras. He starts with two absolutely continuous real-valued functions $$\alpha$$ and $$\beta$$ defined on the interval $$[0,T]$$, where $$\beta$$ is strictly increasing and $$|\alpha|'(t)+\beta'(t) >0$$ for all $$t\in [0,T]$$ and denotes by $$\nu_{\alpha,\beta}$$ the Lebesgue-Stieltjes measure associated to $$|\alpha|+\beta$$ and by $$L^2_{\alpha,\beta}([0,T])$$ the Hilbert space of square integrable functions with respect to $$\nu_{\alpha,\beta}$$.
Following K. S. Ryu [Honam Math. J. 32, No. 4, 633–642 (2010; Zbl 1209.28022)], the author recalls the construction of the analogue of a generalized Wiener measure according to a positive measure $$\phi$$ on the Borel sets of $$\mathbb R$$ which is denoted by $$w_{\alpha,\beta,\phi}$$ and he considers the class $$\mathcal M(B([0,T])$$ of measures of finite variation defined on subsets of $$B([0,T])$$ (real right-continuous functions of bounded variation on $$[0,T]$$ that vanish at $$T$$) whose class of measurable sets is given by $$\{ v\in B([0,T]): \langle v, f\rangle_{\alpha,\beta}<\lambda\}$$ for $$f\in L^2_{\alpha,\beta}([0,T])$$ and $$\lambda\in \mathbb R$$. It is known (see [R. H. Cameron and D. A. Storvick, Lect. Notes Math. 798, 18–67 (1980; Zbl 0439.28007)]) that $$\mathcal M(B([0,T])$$ is a Banach algebra under convolution. He establishes the equivalence classes $$\mu_1\approx \mu_2$$ if $$\int_{B([0,T])} J(x,v)\,d\mu_1(v)=\int_{B([0,T])} J(x,v)\,d\mu_2(v)$$ for $$w_{\alpha,\beta,\phi}$$-almost everywhere $$x\in C([0,T])$$, where $$J(x,v)=\exp\{i \int_0^T v(t)\,dx(t)\}$$.
Denoting by $$\tilde{\mathcal M}(B([0,T])$$ the set of equivalence classes, it is shown in the paper that $$(\tilde{\mathcal M}(B([0,T]), \|\cdot\|)$$ where $$\|[\mu]\|=\inf \{ \|\mu_1\|: \mu_1\approx \mu\}$$, is a Banach algebra with unit. Also, the author defines the space of functions $$F(x)=\int_{B([0,T]} J(x,f)\,d\mu(f)$$ for $$w_{\alpha,\beta,\phi}$$-almost everywhere $$x\in C([0,T])$$ for some $$\mu\in \mathcal M(B([0,T])$$. Defining $$\|F\|=\inf \{\|\mu\|\}$$ where the infimum is taken over all possible $$\mu$$, it is shown that such a space becomes a Banach algebra under pointwise multiplication isometrically isomorphic to $$\tilde{\mathcal M}(B([0,T])$$.

##### MSC:
 46J10 Banach algebras of continuous functions, function algebras 28C20 Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) 60H05 Stochastic integrals 46G12 Measures and integration on abstract linear spaces
##### Citations:
Zbl 1400.46042; Zbl 1209.28022; Zbl 0439.28007
Full Text:
##### References:
 [1] S. A. Albeverio, R. J. H\aoegh-Krohn, and S. Mazzucchi, Mathematical theory of Feynman path integrals, An introduction, 2nd edition, Lecture Notes in Math., 523, Springer-Verlag, Berlin, 2008. · Zbl 1222.46001 [2] R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, Analytic functions, Kozubnik 1979 (Proc. Seventh Conf., Kozubnik, 1979), pp. 18–67, Lecture Notes in Math., 798, Springer, Berlin-New York, 1980. · Zbl 0439.28007 [3] D. H. Cho, Measurable functions similar to the Itô integral and the Paley-Wiener-Zygmund integral over continuous paths, Priprint. [4] D. H. Cho, A Banach algebra similar to the Cameron-Storvick’s one with its equivalent spaces, J. Funct. Spaces (accepted). · Zbl 1400.46042 [5] D. H. Cho, A Banach algebra and its equivalent space over continuous paths with a positive measure, Preprint. [6] R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Rev. Modern Physics 20 (1948), 367–387. · Zbl 1371.81126 [7] G. Kallianpur and C. Bromley, Generalized Feynman integrals using analytic continuation in several complex variables, Stochastic analysis and applications, 217–267, Adv. Probab. Related Topics, 7, Dekker, New York, 1984. · Zbl 0554.60061 [8] I. D. Pierce, On a family of generalized Wiener spaces and applications [Ph.D. thesis], University of Nebraska-Lincoln, Lincoln, Neb, USA, 2011. [9] K. S. Ryu, The generalized analogue of Wiener measure space and its properties, Honam Math. J. 32 (2010), no. 4, 633–642. · Zbl 1209.28022 [10] K. S. Ryu, The translation theorem on the generalized analogue of Wiener space and its applications, J.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.